Spectral Geometry of Nuts and Bolts

TL;DR

This paper analyzes the spectral properties of Laplace operators on a family of gravitational instantons, including Taub-NUT and Schwarzschild geometries, revealing an infinite discrete spectrum and providing numerical and analytical insights.

Contribution

It introduces a unified spectral analysis of a family of geometries interpolating between known gravitational instantons, including handling conical singularities and computing spectra.

Findings

The Laplace operators have self-adjoint extensions despite conical singularities.

The essential spectrum is characterized, and the discrete spectrum is shown to be infinite.

Numerical eigenvalues match well with analytical asymptotic approximations.

Abstract

We study the spectrum of Laplace operators on a one-parameter family of gravitational instantons of bi-axial Bianchi IX type coupled to an abelian connection with self-dual curvature. The family of geometries includes the Taub-NUT, Taub-bolt and Euclidean Schwarzschild geometries and interpolates between them. The interpolating geometries have conical singularities along a submanifold of co-dimension two, but we prove that the associated Laplace operators have natural self-adjoint extensions and study their spectra. In particular, we determine the essential spectrum and prove that its complement, the discrete spectrum, is infinite. We compute these eigenvalues numerically and compare the numerical results with an analytical approximation derived from the asymptotic Taub-NUT form of each of the geometries in our family.